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On boundaries of multiply connected wandering domains

Markus Baumgartner

Christian-Albrechts-Universit¨ at zu Kiel baumgartner@math.uni-kiel.de

London, 12 March 2015

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 1 / 22

Outline

1

Introduction

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 2 / 22

Outline

1

Introduction

2

Results

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 2 / 22

Outline

1

Introduction

2

Results

3

Proof

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 2 / 22

Outline

1

Introduction

2

Results

3

Proof

4

Examples

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 2 / 22

Introduction History

Introduction

Definition (Wandering domain)

Let f be a rational or entire function. A Fatou component U is called wandering domain if f n(U) ∩ f m(U) = ∅ for all m < n.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 3 / 22

Introduction History

Introduction

Definition (Wandering domain)

Let f be a rational or entire function. A Fatou component U is called wandering domain if f n(U) ∩ f m(U) = ∅ for all m < n.

Theorem (Sullivan 1982)

There are no wandering domains for rational functions.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 3 / 22

Introduction History

First example of a wandering domain

The first example of a wandering domain is due to Baker. The function considered was f (z) = C · z2

∞

• j=1
• 1 + z

rj

• ,
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 4 / 22

Introduction History

First example of a wandering domain

The first example of a wandering domain is due to Baker. The function considered was f (z) = C · z2

∞

• j=1
• 1 + z

rj

• ,

where C > 0 is a small constant, r1 is large and (rn)n∈N is a sequence of positive real numbers that satisfies the recurrence relation rn+1 = C · r2

n n

• j=1
• 1 + rn

rj

• .
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 4 / 22

Introduction History

First example of a wandering domain

The first example of a wandering domain is due to Baker. The function considered was f (z) = C · z2

∞

• j=1
• 1 + z

rj

• ,

where C > 0 is a small constant, r1 is large and (rn)n∈N is a sequence of positive real numbers that satisfies the recurrence relation rn+1 = C · r2

n n

• j=1
• 1 + rn

rj

• .

In 1963 Baker showed that f has multiply connected Fatou components Un with f (Un) ⊂ Un+1, but the question whether the Un are all different remained open.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 4 / 22

Introduction History

First example of a wandering domain

The first example of a wandering domain is due to Baker. The function considered was f (z) = C · z2

∞

• j=1
• 1 + z

rj

• ,

where C > 0 is a small constant, r1 is large and (rn)n∈N is a sequence of positive real numbers that satisfies the recurrence relation rn+1 = C · r2

n n

• j=1
• 1 + rn

rj

• .

In 1963 Baker showed that f has multiply connected Fatou components Un with f (Un) ⊂ Un+1, but the question whether the Un are all different remained open. Those were the first known multiply connected Fatou components.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 4 / 22

Introduction History

First example of a wandering domain

The first example of a wandering domain is due to Baker. The function considered was f (z) = C · z2

∞

• j=1
• 1 + z

rj

• ,

where C > 0 is a small constant, r1 is large and (rn)n∈N is a sequence of positive real numbers that satisfies the recurrence relation rn+1 = C · r2

n n

• j=1
• 1 + rn

rj

• .

In 1963 Baker showed that f has multiply connected Fatou components Un with f (Un) ⊂ Un+1, but the question whether the Un are all different remained open. Those were the first known multiply connected Fatou components. In 1976 Baker was able to show that the Un are all different and therefore wandering domains.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 4 / 22

Introduction Shape of a multiply connected wandering domain

rn rn+1 rn+2 f f

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 rn rn+1 rn+2 An An+1 An+2

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 rn rn+1 rn+2 f f f f (Bn) An An+1 An+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An))

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un. Assume that Un = Um for n = m,

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un. Assume that Un = Um for n = m,

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un. Assume that Un = Um for n = m, then this implies that Un = Um for all n, m.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un. Assume that Un = Um for n = m, then this implies that Un = Um for all n, m. Baker showed that there are no unbounded multiply connected Fatou components.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Shape of a multiply connected wandering domain

Bn−1 Bn Bn+1 Bn+2 An An+1 An+2 Un−1 Un Un+1 Un+2 f (Bn) ⊂ Bn+1 (and therefore An+1 ⊂ f (An)) This implies that Bn belongs to a multiply connected Fatou component Un. Assume that Un = Um for n = m, then this implies that Un = Um for all n, m. Baker showed that there are no unbounded multiply connected Fatou components. Baker showed later that every multiply connected wandering domain has similar properties like his first example.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 5 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ).

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ). This implies that J (f ) can not be locally connected at any point if f has a multiply connected wandering domain.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ). This implies that J (f ) can not be locally connected at any point if f has a multiply connected wandering domain.

Question

Are at least the different components of J (f ) locally connected?

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ). This implies that J (f ) can not be locally connected at any point if f has a multiply connected wandering domain.

Question

Are at least the different components of J (f ) locally connected?

Theorem (Bishop 2011)

There exists an entire function f with dimH J (f ) = 1.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ). This implies that J (f ) can not be locally connected at any point if f has a multiply connected wandering domain.

Question

Are at least the different components of J (f ) locally connected?

Theorem (Bishop 2011)

There exists an entire function f with dimH J (f ) = 1. Bishop showed that F(f ) consists of multiply connected wandering domains which are bounded by recitifiable Jordan curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Introduction Motivation

Theorem (Baker and Dominguez 2000)

Let f be an entire function. If J (f ) is not connected, then it is not locally connected at any point of J (f ). This implies that J (f ) can not be locally connected at any point if f has a multiply connected wandering domain.

Question

Are at least the different components of J (f ) locally connected?

Theorem (Bishop 2011)

There exists an entire function f with dimH J (f ) = 1. Bishop showed that F(f ) consists of multiply connected wandering domains which are bounded by recitifiable Jordan curves. We want to show that under suitable conditions every boundary component of a multiply connected wandering domain is a curve or even a Jordan curve and therefore locally connected.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 6 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

U a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

U C(a, U) a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

We call ∂∞U = ∂C(∞, U) the outer boundary component of U and for 0 / ∈ U we call ∂0U = ∂C(0, U) the inner boundary component of U. U C(a, U) a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

We call ∂∞U = ∂C(∞, U) the outer boundary component of U and for 0 / ∈ U we call ∂0U = ∂C(0, U) the inner boundary component of U. U ∂∞U C(a, U) a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

We call ∂∞U = ∂C(∞, U) the outer boundary component of U and for 0 / ∈ U we call ∂0U = ∂C(0, U) the inner boundary component of U. U ∂∞U ∂0U C(a, U) a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Results

Definition (Inner and outer boundary)

Let U ⊂ C be a domain and let a ∈ C \ U. We denote by C(a, U) the component

• f C \ U that contains a.

We call ∂∞U = ∂C(∞, U) the outer boundary component of U and for 0 / ∈ U we call ∂0U = ∂C(0, U) the inner boundary component of U. We call ∂0U and ∂∞U big boundary components. U ∂∞U ∂0U C(a, U) a

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 7 / 22

Results Preparations for the results

Definition (Connectivity)

Let U ⊂ C be a domain. By c(U) we denote the connectivity of U, that is the number of connected components of C \ U.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 8 / 22

Results Preparations for the results

Definition (Connectivity)

Let U ⊂ C be a domain. By c(U) we denote the connectivity of U, that is the number of connected components of C \ U. U

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 8 / 22

Results Preparations for the results

Definition (Connectivity)

Let U ⊂ C be a domain. By c(U) we denote the connectivity of U, that is the number of connected components of C \ U. U c(U) = 6

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 8 / 22

Results Preparations for the results

Definition (Connectivity)

Let U ⊂ C be a domain. By c(U) we denote the connectivity of U, that is the number of connected components of C \ U. For a sequence of domains Un we call c the eventual connectivity of Un if c(Un) = c for all large n. U c(U) = 6

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 8 / 22

Results Preparations for the results

Definition (Connectivity)

Let U ⊂ C be a domain. By c(U) we denote the connectivity of U, that is the number of connected components of C \ U. For a sequence of domains Un we call c the eventual connectivity of Un if c(Un) = c for all large n. U c(U) = 6 Kisaka and Shishikura showed that the eventual connectivity of a multiply connected wandering domain is either 2 or ∞.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 8 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U).

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m K f m(K)

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un. We define the eventual inner and outer connectivity respectively.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un. We define the eventual inner and outer connectivity respectively.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un. We define the eventual inner and outer connectivity respectively.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Preparations for the results

Theorem (Bergweiler, Rippon, Stallard 2013)

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Then every Un contains an annulus Bn such that every compact subset K ⊂ Un is mapped inside Bn+m under f m for all large m ∈ N. Un Un+m Bn Bn+m

Definition (Inner and outer connectivity)

We call c(Un ∩ C(0, Bn)) the inner connectivity and c(Un ∩ C(∞, Bn)) the outer connectivity of Un. We define the eventual inner and outer connectivity respectively. BRS showed that the eventual inner and outer connectivity is also either 2 or ∞.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 9 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U).

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Suppose that there exists a sequence of positive real numbers (rn)n∈N as well as α, β > 0 such that for a sequence of annuli Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} the following conditions hold:

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Suppose that there exists a sequence of positive real numbers (rn)n∈N as well as α, β > 0 such that for a sequence of annuli Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} the following conditions hold: ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Suppose that there exists a sequence of positive real numbers (rn)n∈N as well as α, β > 0 such that for a sequence of annuli Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} the following conditions hold: ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un Cn+1 ⊂ f (Cn)

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Suppose that there exists a sequence of positive real numbers (rn)n∈N as well as α, β > 0 such that for a sequence of annuli Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} the following conditions hold: ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un Cn+1 ⊂ f (Cn) There exists m > β

α such that for all z ∈ f −1(Cn+1) ∩ Cn

• z · f ′(z)

f (z)

• ≥ m

and

• arg

z · f ′(z) f (z)

• < π

2 .

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Main result

Theorem 1

Let f be an entire function with a multiply connected wandering domain U = U0. Denote Un = f n(U). Suppose that there exists a sequence of positive real numbers (rn)n∈N as well as α, β > 0 such that for a sequence of annuli Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} the following conditions hold: ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un Cn+1 ⊂ f (Cn) There exists m > β

α such that for all z ∈ f −1(Cn+1) ∩ Cn

• z · f ′(z)

f (z)

• ≥ m

and

• arg

z · f ′(z) f (z)

• < π

2 . Then all big boundary components are Jordan curves and ∂∞Un−1 = ∂0Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 10 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn. Then all big boundary components are rectifiable Jordan curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn. Then all big boundary components are rectifiable Jordan curves.

Definition (Eventually big boundary components)

Let f be an entire function with a multiply connected wandering domain U. Let Z be a boundary component of U.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn. Then all big boundary components are rectifiable Jordan curves.

Definition (Eventually big boundary components)

Let f be an entire function with a multiply connected wandering domain U. Let Z be a boundary component of U. We call Z eventually big if f n(Z) is a big boundary component of Un for some n ∈ N.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn. Then all big boundary components are rectifiable Jordan curves.

Definition (Eventually big boundary components)

Let f be an entire function with a multiply connected wandering domain U. Let Z be a boundary component of U. We call Z eventually big if f n(Z) is a big boundary component of Un for some n ∈ N.

Corollary 1

Let Z be an eventually big boundary component of U.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Theorem 2

Let (εn)n∈N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have

• arg

z · f ′(z) f (z)

• < εn

for all z ∈ f −1(Cn+1) ∩ Cn. Then all big boundary components are rectifiable Jordan curves.

Definition (Eventually big boundary components)

Let f be an entire function with a multiply connected wandering domain U. Let Z be a boundary component of U. We call Z eventually big if f n(Z) is a big boundary component of Un for some n ∈ N.

Corollary 1

Let Z be an eventually big boundary component of U. Then Z is a closed (rectifiable) curve. Moreover Z is a (rectifiable) Jordan curve if f j(Z) does not contain any critical points for all j ∈ N0.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 11 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U. The eventual inner connectivity of U is 2 if and only if every boundary component

• f U is eventually big.
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U. The eventual inner connectivity of U is 2 if and only if every boundary component

• f U is eventually big.

One direction of the lemma together with corollary 1 implies the following corollary:

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U. The eventual inner connectivity of U is 2 if and only if every boundary component

• f U is eventually big.

One direction of the lemma together with corollary 1 implies the following corollary:

Corollary 2

Suppose that the eventual inner connectivity of Un is 2.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U. The eventual inner connectivity of U is 2 if and only if every boundary component

• f U is eventually big.

One direction of the lemma together with corollary 1 implies the following corollary:

Corollary 2

Suppose that the eventual inner connectivity of Un is 2. Then all wandering domains, which belong to the orbit of Un, are bounded by a countable number of closed (rectifiable) curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Results Further results

Lemma (Joint work with Rippon and Stallard)

Let f be an entire function with a multiply connected wandering domain U. The eventual inner connectivity of U is 2 if and only if every boundary component

• f U is eventually big.

One direction of the lemma together with corollary 1 implies the following corollary:

Corollary 2

Suppose that the eventual inner connectivity of Un is 2. Then all wandering domains, which belong to the orbit of Un, are bounded by a countable number of closed (rectifiable) curves. We can apply Theorem 1 and both corollaries for Baker’s first example of a wandering domain. This means that every multiply connected wandering domain in Baker’s first example is bounded by a countable number of Jordan curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 12 / 22

Proof Idea of the proof of theorem 1

Proof

Understanding the setting of theorem 1:

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 13 / 22

Proof Idea of the proof of theorem 1

Proof

Understanding the setting of theorem 1: Cn Cn+1 Cn+2 rn βrn αrn Cn := {z ∈ C : αrn ≤ |z| ≤ βrn}

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 13 / 22

Proof Idea of the proof of theorem 1

Proof

Understanding the setting of theorem 1: Cn Cn+1 Cn+2 rn βrn αrn Un−1 Un Un+1 Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 13 / 22

Proof Idea of the proof of theorem 1

Proof

Understanding the setting of theorem 1: Cn Cn+1 Cn+2 rn βrn αrn Un−1 Un Un+1 f Cn := {z ∈ C : αrn ≤ |z| ≤ βrn} ∂0Cn ⊂ Un−1, ∂∞Cn ⊂ Un Cn+1 ⊂ f (Cn)

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 13 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}. The inequality

• z·f ′(z)

f (z)

• ≥ m implies that there are no critical points inside the Γk.

So by the Riemann-Hurwitz-formula all Γk are topological annuli that are bounded by Jordan curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}. The inequality

• z·f ′(z)

f (z)

• ≥ m implies that there are no critical points inside the Γk.

So by the Riemann-Hurwitz-formula all Γk are topological annuli that are bounded by Jordan curves. Cn Cn+1 Cn+2 Cn+3

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}. The inequality

• z·f ′(z)

f (z)

• ≥ m implies that there are no critical points inside the Γk.

So by the Riemann-Hurwitz-formula all Γk are topological annuli that are bounded by Jordan curves. Γ1 Cn Cn+1 Cn+2 Cn+3 f

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}. The inequality

• z·f ′(z)

f (z)

• ≥ m implies that there are no critical points inside the Γk.

So by the Riemann-Hurwitz-formula all Γk are topological annuli that are bounded by Jordan curves. Γ1 Γ2 Cn Cn+1 Cn+2 Cn+3 f f 2

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

We want to show that ∂∞Un−1 and ∂0Un are both curves that coincide. Define for all k ∈ N Γk := {z ∈ Cn : f j(z) ∈ Cn+j for all j=1,. . . ,k}. The inequality

• z·f ′(z)

f (z)

• ≥ m implies that there are no critical points inside the Γk.

So by the Riemann-Hurwitz-formula all Γk are topological annuli that are bounded by Jordan curves. Γ1 Γ3 Γ2 Cn Cn+1 Cn+2 Cn+3 f f 2 f 3

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 14 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:

Lemma

There exists ̺ > 1 such that we have

• f k′ (z)
• ≥ ̺k · rn+k

rn for all k ∈ N and z ∈ Γk.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:

Lemma

There exists ̺ > 1 such that we have

• f k′ (z)
• ≥ ̺k · rn+k

rn for all k ∈ N and z ∈ Γk. Therefore, f k is expanding inside Γk and this implies that f −k : Cn+k → Γk is contracting.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:

Lemma

There exists ̺ > 1 such that we have

• f k′ (z)
• ≥ ̺k · rn+k

rn for all k ∈ N and z ∈ Γk. Therefore, f k is expanding inside Γk and this implies that f −k : Cn+k → Γk is contracting. We parametrise now ∂0Γk and ∂∞Γk as curves by γ0

k and γ∞ k

respectively.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:

Lemma

There exists ̺ > 1 such that we have

• f k′ (z)
• ≥ ̺k · rn+k

rn for all k ∈ N and z ∈ Γk. Therefore, f k is expanding inside Γk and this implies that f −k : Cn+k → Γk is contracting. We parametrise now ∂0Γk and ∂∞Γk as curves by γ0

k and γ∞ k

respectively. Thereby, one has to check that the parametrisations are compatible with each

• ther.
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

The inequality

• z·f ′(z)

f (z)

• ≥ m also implies the following lemma:

Lemma

There exists ̺ > 1 such that we have

• f k′ (z)
• ≥ ̺k · rn+k

rn for all k ∈ N and z ∈ Γk. Therefore, f k is expanding inside Γk and this implies that f −k : Cn+k → Γk is contracting. We parametrise now ∂0Γk and ∂∞Γk as curves by γ0

k and γ∞ k

respectively. Thereby, one has to check that the parametrisations are compatible with each

• ther. Here
• arg
• z·f ′(z)

f (z)

• < π

2 ensures that the curves are not distorted too much

under iteration.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 15 / 22

Proof Idea of the proof of theorem 1

∂0Un ∂∞Un−1 ∂0Cn ∂∞Cn

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 16 / 22

Proof Idea of the proof of theorem 1

∂0Un ∂∞Un−1 ∂0Cn ∂∞Cn γ0

1

γ∞

1

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 16 / 22

Proof Idea of the proof of theorem 1

∂0Un ∂∞Un−1 ∂0Cn ∂∞Cn γ0

2

γ0

1

γ∞

1

γ∞

2

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 16 / 22

Proof Idea of the proof of theorem 1

∂0Un ∂∞Un−1 ∂0Cn ∂∞Cn γ0

3

γ0

2

γ0

1

γ∞

1

γ∞

3

γ∞

2

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 16 / 22

Proof Idea of the proof of theorem 1

γ ∂0Un ∂∞Un−1 ∂0Cn ∂∞Cn γ0

3

γ0

2

γ0

1

γ∞

1

γ∞

3

γ∞

2

Then we use that f −k is contracting to show that the curves γ0

k and γ∞ k

converge uniformly to the same curve γ with trace(γ) =

• k∈N

Γk.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 16 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1. In the proof of theorem 1 we used that

• arg
• z·f ′(z)

f (z)

• < π

2 bounds the distortion

• f the curves under iteration.
• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1. In the proof of theorem 1 we used that

• arg
• z·f ′(z)

f (z)

• < π

2 bounds the distortion

• f the curves under iteration.

In order to prove theorem 2 we exploit that

• arg
• z·f ′(z)

f (z)

• < εn ensures that the

curves are only distorted by a very small amount under iteration.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1. In the proof of theorem 1 we used that

• arg
• z·f ′(z)

f (z)

• < π

2 bounds the distortion

• f the curves under iteration.

In order to prove theorem 2 we exploit that

• arg
• z·f ′(z)

f (z)

• < εn ensures that the

curves are only distorted by a very small amount under iteration. Thus we are able to show that the curves are close to circles.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1. In the proof of theorem 1 we used that

• arg
• z·f ′(z)

f (z)

• < π

2 bounds the distortion

• f the curves under iteration.

In order to prove theorem 2 we exploit that

• arg
• z·f ′(z)

f (z)

• < εn ensures that the

curves are only distorted by a very small amount under iteration. Thus we are able to show that the curves are close to circles. Therefore, γ is itself close to a circle and hence rectifiable.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of theorem 1 and 2

By positioning of Cn to Un−1 and Un we have ∂∞Un−1 = trace(γ) = ∂0Un. Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂∞Un−1 and ∂0Un are curves and therefore locally connected, every point

• n trace(γ) is accessible in Un−1 and in Un.

Thus a theorem of Sch¨

• nflies yields that γ is in fact a Jordan curve.

This proves theorem 1. In the proof of theorem 1 we used that

• arg
• z·f ′(z)

f (z)

• < π

2 bounds the distortion

• f the curves under iteration.

In order to prove theorem 2 we exploit that

• arg
• z·f ′(z)

f (z)

• < εn ensures that the

curves are only distorted by a very small amount under iteration. Thus we are able to show that the curves are close to circles. Therefore, γ is itself close to a circle and hence rectifiable. This proves theorem 2.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 17 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components. Suppose Z is a boundary component of U which is not a big boundary component.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components. Suppose Z is a boundary component of U which is not a big boundary component. U Un Z γ

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components. Suppose Z is a boundary component of U which is not a big boundary component. Bn U Un Z γ This proves the nedded direction of the lemma.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components. Suppose Z is a boundary component of U which is not a big boundary component. Bn U Un Z γ f n(γ) This proves the nedded direction of the lemma.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Proof Idea of the proof of the lemma

Now we want to prove that if the eventual inner connectivity of U is 2 every boundary component of U will be eventually mapped onto a big boundary component. By the maximum and minimum modulus principle it is clear that big boundary components are mapped onto big boundary components. Suppose Z is a boundary component of U which is not a big boundary component. Bn U Un Z f n(Z) γ f n(γ) This proves the nedded direction of the lemma.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 18 / 22

Examples Bergweiler’s and Zheng’s example

Examples

In the following we are looking at three different classes of entire functions with multiply connected wandering domains to which we can apply the theorems.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 19 / 22

Examples Bergweiler’s and Zheng’s example

Examples

In the following we are looking at three different classes of entire functions with multiply connected wandering domains to which we can apply the theorems.

Bergweiler’s and Zheng’s example

f (z) = C · zk

∞

• n=1
• 1 − z

an

• ,

where C > 0, k ∈ N and (an)n∈N is a complex sequence with |an| = rn and (rn)n∈N is a fast growing sequence of positive real numbers.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 19 / 22

Examples Bergweiler’s and Zheng’s example

Examples

In the following we are looking at three different classes of entire functions with multiply connected wandering domains to which we can apply the theorems.

Bergweiler’s and Zheng’s example

f (z) = C · zk

∞

• n=1
• 1 − z

an

• ,

where C > 0, k ∈ N and (an)n∈N is a complex sequence with |an| = rn and (rn)n∈N is a fast growing sequence of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 1 hold.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 19 / 22

Examples Bergweiler’s and Zheng’s example

Examples

In the following we are looking at three different classes of entire functions with multiply connected wandering domains to which we can apply the theorems.

Bergweiler’s and Zheng’s example

f (z) = C · zk

∞

• n=1
• 1 − z

an

• ,

where C > 0, k ∈ N and (an)n∈N is a complex sequence with |an| = rn and (rn)n∈N is a fast growing sequence of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 1 hold. This example includes the first example of Baker. In this case the eventual inner connectivity is 2.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 19 / 22

Examples Baker’s infinite connectivity example

Baker’s infinite connectivity example

f (z) = C ·

∞

• n=1
• 1 + z

rn k , where C > 0, k ∈ N and (rn)n∈N is a fast growing sequence of positive real numbers.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 20 / 22

Examples Baker’s infinite connectivity example

Baker’s infinite connectivity example

f (z) = C ·

∞

• n=1
• 1 + z

rn k , where C > 0, k ∈ N and (rn)n∈N is a fast growing sequence of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 1 hold.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 20 / 22

Examples Baker’s infinite connectivity example

Baker’s infinite connectivity example

f (z) = C ·

∞

• n=1
• 1 + z

rn k , where C > 0, k ∈ N and (rn)n∈N is a fast growing sequence of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 1 hold. This example includes the first example of Baker (1984) with a wandering domain with infinite connectivity. In this case the eventual inner connectivity is 2.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 20 / 22

Examples Baker’s infinite connectivity example

Baker’s infinite connectivity example

f (z) = C ·

∞

• n=1
• 1 + z

rn k , where C > 0, k ∈ N and (rn)n∈N is a fast growing sequence of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 1 hold. This example includes the first example of Baker (1984) with a wandering domain with infinite connectivity. In this case the eventual inner connectivity is 2. Bergweiler and Zheng showed that Baker’s first example of a wandering domain has also infinite connectivity.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 20 / 22

Examples Baker’s example of arbitrary order

Baker’s example of arbitrary order

f (z) = C ·

∞

• n=1
• 1 +

z rn kn , where C > 0 and (rn)n∈N, (kn)n∈N are fast growing sequences of positive real numbers.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 21 / 22

Examples Baker’s example of arbitrary order

Baker’s example of arbitrary order

f (z) = C ·

∞

• n=1
• 1 +

z rn kn , where C > 0 and (rn)n∈N, (kn)n∈N are fast growing sequences of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 2 hold.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 21 / 22

Examples Baker’s example of arbitrary order

Baker’s example of arbitrary order

f (z) = C ·

∞

• n=1
• 1 +

z rn kn , where C > 0 and (rn)n∈N, (kn)n∈N are fast growing sequences of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 2 hold. This example includes the first example of Baker (1984) with arbitrary order.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 21 / 22

Examples Baker’s example of arbitrary order

Baker’s example of arbitrary order

f (z) = C ·

∞

• n=1
• 1 +

z rn kn , where C > 0 and (rn)n∈N, (kn)n∈N are fast growing sequences of positive real numbers. The sequence (rn)n∈N can be chosen such that the conditions of theorem 2 hold. This example includes the first example of Baker (1984) with arbitrary order. Bishop’s example which was the starting point of my work is also constructed by an infinite product which is similar to the one above.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 21 / 22

The End

Thank you for your attention.

• M. Baumgartner (University of Kiel)

Boundaries of wandering domains London, 12 March 2015 22 / 22